The \( {\mathbbm{CP}}^{N-1} \) model is an analytically tractable 2d quantum field theory which shares several properties with 4d Yang-Mills theory. By virtue of its classical integrability, this model also admits a family of integrable higher-spin auxiliary field deformations, including the \( T\overline{T} \) deformation as a special case. We study the \( {\mathbbm{CP}}^{N-1} \) model and its deformations from a geometrical perspective, constructing their soliton surfaces and recasting physical properties of these theories as statements about surface geometry. We examine how the \( T\overline{T} \) flow affects the unit constraint in the \( {\mathbbm{CP}}^{N-1} \) model and prove that any solution of this theory with vanishing energy-momentum tensor remains a solution under analytic stress tensor deformations — an argument that extends to generic dimensions and instanton-like solutions in stress tensor flows including the non-analytic, 2d, root- \( T\overline{T} \) case and classes of higher-spin, Smirnov-Zamolodchikov-type, deformations. Finally, we give two geometric interpretations for general \( T\overline{T} \) -like deformations of symmetric space sigma models, showing that such flows can be viewed as coupling the undeformed theory to a unit-determinant field-dependent metric, or using a particular choice of moving frame on the soliton surface.
